“Either way, the blade is no longer considered Anakin’s lightsaber by the Lucasfilm marketing department.
From there, it seems that Luke would rather keep his own-built green lightsaber and willingly let Rey use Anakin’s.
To say that the decision to officially refer to the lightsaber as Rey’s instead of Anakin’s (or maybe even Luke’s) intriguing is an understatement.
LucasFilm has just confirmed that starting Star Wars: Episode VIII – The Last Jedi and moving forward, it will be the main weapon of choice for the former scavenger.